Theorem xlt2add 10261

Description: Extended real version of lt2add 8763. Note that ltleadd 8764, which has weaker assumptions, is not true for the extended reals (since 0 + +oo < 1 + +oo fails). (Contributed by Mario Carneiro, 23-Aug-2015.)

Assertion

Ref	Expression
xlt2add	|- ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) ) -> ( ( A < C /\ B < D ) -> ( A +e B ) < ( C +e D ) ) )

Proof of Theorem xlt2add

Step	Hyp	Ref	Expression
1		xaddcl 10241	. . . . . . . 8 |- ( ( A e. RR* /\ B e. RR* ) -> ( A +e B ) e. RR* )
2	1	3ad2ant1 1049	. . . . . . 7 |- ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) /\ ( A < C /\ B < D ) ) -> ( A +e B ) e. RR* )
3	2	adantr 276	. . . . . 6 |- ( ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) /\ ( A < C /\ B < D ) ) /\ ( A e. RR /\ D e. RR ) ) -> ( A +e B ) e. RR* )
4		simp1l 1052	. . . . . . . 8 |- ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) /\ ( A < C /\ B < D ) ) -> A e. RR* )
5		simp2r 1055	. . . . . . . 8 |- ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) /\ ( A < C /\ B < D ) ) -> D e. RR* )
6		xaddcl 10241	. . . . . . . 8 |- ( ( A e. RR* /\ D e. RR* ) -> ( A +e D ) e. RR* )
7	4, 5, 6	syl2anc 415	. . . . . . 7 |- ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) /\ ( A < C /\ B < D ) ) -> ( A +e D ) e. RR* )
8	7	adantr 276	. . . . . 6 |- ( ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) /\ ( A < C /\ B < D ) ) /\ ( A e. RR /\ D e. RR ) ) -> ( A +e D ) e. RR* )
9		xaddcl 10241	. . . . . . . 8 |- ( ( C e. RR* /\ D e. RR* ) -> ( C +e D ) e. RR* )
10	9	3ad2ant2 1050	. . . . . . 7 |- ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) /\ ( A < C /\ B < D ) ) -> ( C +e D ) e. RR* )
11	10	adantr 276	. . . . . 6 |- ( ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) /\ ( A < C /\ B < D ) ) /\ ( A e. RR /\ D e. RR ) ) -> ( C +e D ) e. RR* )
12		simp3r 1057	. . . . . . . 8 |- ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) /\ ( A < C /\ B < D ) ) -> B < D )
13	12	adantr 276	. . . . . . 7 |- ( ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) /\ ( A < C /\ B < D ) ) /\ ( A e. RR /\ D e. RR ) ) -> B < D )
14		simp1r 1053	. . . . . . . . 9 |- ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) /\ ( A < C /\ B < D ) ) -> B e. RR* )
15	14	adantr 276	. . . . . . . 8 |- ( ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) /\ ( A < C /\ B < D ) ) /\ ( A e. RR /\ D e. RR ) ) -> B e. RR* )
16	5	adantr 276	. . . . . . . 8 |- ( ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) /\ ( A < C /\ B < D ) ) /\ ( A e. RR /\ D e. RR ) ) -> D e. RR* )
17		simprl 535	. . . . . . . 8 |- ( ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) /\ ( A < C /\ B < D ) ) /\ ( A e. RR /\ D e. RR ) ) -> A e. RR )
18		xltadd2 10258	. . . . . . . 8 |- ( ( B e. RR* /\ D e. RR* /\ A e. RR ) -> ( B < D <-> ( A +e B ) < ( A +e D ) ) )
19	15, 16, 17, 18	syl3anc 1278	. . . . . . 7 |- ( ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) /\ ( A < C /\ B < D ) ) /\ ( A e. RR /\ D e. RR ) ) -> ( B < D <-> ( A +e B ) < ( A +e D ) ) )
20	13, 19	mpbid 147	. . . . . 6 |- ( ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) /\ ( A < C /\ B < D ) ) /\ ( A e. RR /\ D e. RR ) ) -> ( A +e B ) < ( A +e D ) )
21		simp3l 1056	. . . . . . . 8 |- ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) /\ ( A < C /\ B < D ) ) -> A < C )
22	21	adantr 276	. . . . . . 7 |- ( ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) /\ ( A < C /\ B < D ) ) /\ ( A e. RR /\ D e. RR ) ) -> A < C )
23	4	adantr 276	. . . . . . . 8 |- ( ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) /\ ( A < C /\ B < D ) ) /\ ( A e. RR /\ D e. RR ) ) -> A e. RR* )
24		simp2l 1054	. . . . . . . . 9 |- ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) /\ ( A < C /\ B < D ) ) -> C e. RR* )
25	24	adantr 276	. . . . . . . 8 |- ( ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) /\ ( A < C /\ B < D ) ) /\ ( A e. RR /\ D e. RR ) ) -> C e. RR* )
26		simprr 537	. . . . . . . 8 |- ( ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) /\ ( A < C /\ B < D ) ) /\ ( A e. RR /\ D e. RR ) ) -> D e. RR )
27		xltadd1 10257	. . . . . . . 8 |- ( ( A e. RR* /\ C e. RR* /\ D e. RR ) -> ( A < C <-> ( A +e D ) < ( C +e D ) ) )
28	23, 25, 26, 27	syl3anc 1278	. . . . . . 7 |- ( ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) /\ ( A < C /\ B < D ) ) /\ ( A e. RR /\ D e. RR ) ) -> ( A < C <-> ( A +e D ) < ( C +e D ) ) )
29	22, 28	mpbid 147	. . . . . 6 |- ( ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) /\ ( A < C /\ B < D ) ) /\ ( A e. RR /\ D e. RR ) ) -> ( A +e D ) < ( C +e D ) )
30	3, 8, 11, 20, 29	xrlttrd 10190	. . . . 5 |- ( ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) /\ ( A < C /\ B < D ) ) /\ ( A e. RR /\ D e. RR ) ) -> ( A +e B ) < ( C +e D ) )
31	30	anassrs 404	. . . 4 |- ( ( ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) /\ ( A < C /\ B < D ) ) /\ A e. RR ) /\ D e. RR ) -> ( A +e B ) < ( C +e D ) )
32		pnfxr 8368	. . . . . . . . . . . 12 |- +oo e. RR*
33	32	a1i 9	. . . . . . . . . . 11 |- ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) /\ ( A < C /\ B < D ) ) -> +oo e. RR* )
34		pnfge 10170	. . . . . . . . . . . 12 |- ( C e. RR* -> C <_ +oo )
35	24, 34	syl 14	. . . . . . . . . . 11 |- ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) /\ ( A < C /\ B < D ) ) -> C <_ +oo )
36	4, 24, 33, 21, 35	xrltletrd 10192	. . . . . . . . . 10 |- ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) /\ ( A < C /\ B < D ) ) -> A < +oo )
37		npnflt 10196	. . . . . . . . . . 11 |- ( A e. RR* -> ( A < +oo <-> A =/= +oo ) )
38	4, 37	syl 14	. . . . . . . . . 10 |- ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) /\ ( A < C /\ B < D ) ) -> ( A < +oo <-> A =/= +oo ) )
39	36, 38	mpbid 147	. . . . . . . . 9 |- ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) /\ ( A < C /\ B < D ) ) -> A =/= +oo )
40		pnfge 10170	. . . . . . . . . . . 12 |- ( D e. RR* -> D <_ +oo )
41	5, 40	syl 14	. . . . . . . . . . 11 |- ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) /\ ( A < C /\ B < D ) ) -> D <_ +oo )
42	14, 5, 33, 12, 41	xrltletrd 10192	. . . . . . . . . 10 |- ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) /\ ( A < C /\ B < D ) ) -> B < +oo )
43		npnflt 10196	. . . . . . . . . . 11 |- ( B e. RR* -> ( B < +oo <-> B =/= +oo ) )
44	14, 43	syl 14	. . . . . . . . . 10 |- ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) /\ ( A < C /\ B < D ) ) -> ( B < +oo <-> B =/= +oo ) )
45	42, 44	mpbid 147	. . . . . . . . 9 |- ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) /\ ( A < C /\ B < D ) ) -> B =/= +oo )
46		xaddnepnf 10239	. . . . . . . . 9 |- ( ( ( A e. RR* /\ A =/= +oo ) /\ ( B e. RR* /\ B =/= +oo ) ) -> ( A +e B ) =/= +oo )
47	4, 39, 14, 45, 46	syl22anc 1279	. . . . . . . 8 |- ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) /\ ( A < C /\ B < D ) ) -> ( A +e B ) =/= +oo )
48		npnflt 10196	. . . . . . . . 9 |- ( ( A +e B ) e. RR* -> ( ( A +e B ) < +oo <-> ( A +e B ) =/= +oo ) )
49	2, 48	syl 14	. . . . . . . 8 |- ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) /\ ( A < C /\ B < D ) ) -> ( ( A +e B ) < +oo <-> ( A +e B ) =/= +oo ) )
50	47, 49	mpbird 167	. . . . . . 7 |- ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) /\ ( A < C /\ B < D ) ) -> ( A +e B ) < +oo )
51	50	adantr 276	. . . . . 6 |- ( ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) /\ ( A < C /\ B < D ) ) /\ D = +oo ) -> ( A +e B ) < +oo )
52		oveq2 6083	. . . . . . 7 |- ( D = +oo -> ( C +e D ) = ( C +e +oo ) )
53		mnfxr 8372	. . . . . . . . . . 11 |- -oo e. RR*
54	53	a1i 9	. . . . . . . . . 10 |- ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) /\ ( A < C /\ B < D ) ) -> -oo e. RR* )
55		mnfle 10173	. . . . . . . . . . 11 |- ( A e. RR* -> -oo <_ A )
56	4, 55	syl 14	. . . . . . . . . 10 |- ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) /\ ( A < C /\ B < D ) ) -> -oo <_ A )
57	54, 4, 24, 56, 21	xrlelttrd 10191	. . . . . . . . 9 |- ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) /\ ( A < C /\ B < D ) ) -> -oo < C )
58		nmnfgt 10199	. . . . . . . . . 10 |- ( C e. RR* -> ( -oo < C <-> C =/= -oo ) )
59	24, 58	syl 14	. . . . . . . . 9 |- ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) /\ ( A < C /\ B < D ) ) -> ( -oo < C <-> C =/= -oo ) )
60	57, 59	mpbid 147	. . . . . . . 8 |- ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) /\ ( A < C /\ B < D ) ) -> C =/= -oo )
61		xaddpnf1 10227	. . . . . . . 8 |- ( ( C e. RR* /\ C =/= -oo ) -> ( C +e +oo ) = +oo )
62	24, 60, 61	syl2anc 415	. . . . . . 7 |- ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) /\ ( A < C /\ B < D ) ) -> ( C +e +oo ) = +oo )
63	52, 62	sylan9eqr 2293	. . . . . 6 |- ( ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) /\ ( A < C /\ B < D ) ) /\ D = +oo ) -> ( C +e D ) = +oo )
64	51, 63	breqtrrd 4153	. . . . 5 |- ( ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) /\ ( A < C /\ B < D ) ) /\ D = +oo ) -> ( A +e B ) < ( C +e D ) )
65	64	adantlr 481	. . . 4 |- ( ( ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) /\ ( A < C /\ B < D ) ) /\ A e. RR ) /\ D = +oo ) -> ( A +e B ) < ( C +e D ) )
66		simpr 110	. . . . . 6 |- ( ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) /\ ( A < C /\ B < D ) ) /\ D = -oo ) -> D = -oo )
67		mnfle 10173	. . . . . . . . . 10 |- ( B e. RR* -> -oo <_ B )
68	14, 67	syl 14	. . . . . . . . 9 |- ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) /\ ( A < C /\ B < D ) ) -> -oo <_ B )
69	54, 14, 5, 68, 12	xrlelttrd 10191	. . . . . . . 8 |- ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) /\ ( A < C /\ B < D ) ) -> -oo < D )
70		nmnfgt 10199	. . . . . . . . 9 |- ( D e. RR* -> ( -oo < D <-> D =/= -oo ) )
71	5, 70	syl 14	. . . . . . . 8 |- ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) /\ ( A < C /\ B < D ) ) -> ( -oo < D <-> D =/= -oo ) )
72	69, 71	mpbid 147	. . . . . . 7 |- ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) /\ ( A < C /\ B < D ) ) -> D =/= -oo )
73	72	adantr 276	. . . . . 6 |- ( ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) /\ ( A < C /\ B < D ) ) /\ D = -oo ) -> D =/= -oo )
74	66, 73	pm2.21ddne 2503	. . . . 5 |- ( ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) /\ ( A < C /\ B < D ) ) /\ D = -oo ) -> ( A +e B ) < ( C +e D ) )
75	74	adantlr 481	. . . 4 |- ( ( ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) /\ ( A < C /\ B < D ) ) /\ A e. RR ) /\ D = -oo ) -> ( A +e B ) < ( C +e D ) )
76		elxr 10157	. . . . . 6 |- ( D e. RR* <-> ( D e. RR \/ D = +oo \/ D = -oo ) )
77	5, 76	sylib 122	. . . . 5 |- ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) /\ ( A < C /\ B < D ) ) -> ( D e. RR \/ D = +oo \/ D = -oo ) )
78	77	adantr 276	. . . 4 |- ( ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) /\ ( A < C /\ B < D ) ) /\ A e. RR ) -> ( D e. RR \/ D = +oo \/ D = -oo ) )
79	31, 65, 75, 78	mpjao3dan 1348	. . 3 |- ( ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) /\ ( A < C /\ B < D ) ) /\ A e. RR ) -> ( A +e B ) < ( C +e D ) )
80		simpr 110	. . . 4 |- ( ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) /\ ( A < C /\ B < D ) ) /\ A = +oo ) -> A = +oo )
81	39	adantr 276	. . . 4 |- ( ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) /\ ( A < C /\ B < D ) ) /\ A = +oo ) -> A =/= +oo )
82	80, 81	pm2.21ddne 2503	. . 3 |- ( ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) /\ ( A < C /\ B < D ) ) /\ A = +oo ) -> ( A +e B ) < ( C +e D ) )
83		oveq1 6082	. . . . 5 |- ( A = -oo -> ( A +e B ) = ( -oo +e B ) )
84		xaddmnf2 10230	. . . . . 6 |- ( ( B e. RR* /\ B =/= +oo ) -> ( -oo +e B ) = -oo )
85	14, 45, 84	syl2anc 415	. . . . 5 |- ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) /\ ( A < C /\ B < D ) ) -> ( -oo +e B ) = -oo )
86	83, 85	sylan9eqr 2293	. . . 4 |- ( ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) /\ ( A < C /\ B < D ) ) /\ A = -oo ) -> ( A +e B ) = -oo )
87		xaddnemnf 10238	. . . . . . 7 |- ( ( ( C e. RR* /\ C =/= -oo ) /\ ( D e. RR* /\ D =/= -oo ) ) -> ( C +e D ) =/= -oo )
88	24, 60, 5, 72, 87	syl22anc 1279	. . . . . 6 |- ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) /\ ( A < C /\ B < D ) ) -> ( C +e D ) =/= -oo )
89		nmnfgt 10199	. . . . . . 7 |- ( ( C +e D ) e. RR* -> ( -oo < ( C +e D ) <-> ( C +e D ) =/= -oo ) )
90	10, 89	syl 14	. . . . . 6 |- ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) /\ ( A < C /\ B < D ) ) -> ( -oo < ( C +e D ) <-> ( C +e D ) =/= -oo ) )
91	88, 90	mpbird 167	. . . . 5 |- ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) /\ ( A < C /\ B < D ) ) -> -oo < ( C +e D ) )
92	91	adantr 276	. . . 4 |- ( ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) /\ ( A < C /\ B < D ) ) /\ A = -oo ) -> -oo < ( C +e D ) )
93	86, 92	eqbrtrd 4147	. . 3 |- ( ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) /\ ( A < C /\ B < D ) ) /\ A = -oo ) -> ( A +e B ) < ( C +e D ) )
94		elxr 10157	. . . 4 |- ( A e. RR* <-> ( A e. RR \/ A = +oo \/ A = -oo ) )
95	4, 94	sylib 122	. . 3 |- ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) /\ ( A < C /\ B < D ) ) -> ( A e. RR \/ A = +oo \/ A = -oo ) )
96	79, 82, 93, 95	mpjao3dan 1348	. 2 |- ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) /\ ( A < C /\ B < D ) ) -> ( A +e B ) < ( C +e D ) )
97	96	3expia 1236	1 |- ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) ) -> ( ( A < C /\ B < D ) -> ( A +e B ) < ( C +e D ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   \/ w3o 1008   /\ w3a 1009   = wceq 1402   e. wcel 2209   =/= wne 2420   class class class wbr 4125  (class class class)co 6075   RRcr 8168   +oocpnf 8347   -oocmnf 8348   RR*cxr 8349   < clt 8350   <_ cle 8351   +ecxad 10151

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679  ax-cnex 8260  ax-resscn 8261  ax-1cn 8262  ax-1re 8263  ax-icn 8264  ax-addcl 8265  ax-addrcl 8266  ax-mulcl 8267  ax-addcom 8269  ax-addass 8271  ax-i2m1 8274  ax-0id 8277  ax-rnegex 8278  ax-pre-ltirr 8281  ax-pre-ltwlin 8282  ax-pre-lttrn 8283  ax-pre-ltadd 8285

This theorem depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-if 3636  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-iun 4009  df-br 4126  df-opab 4188  df-mpt 4189  df-id 4433  df-po 4436  df-iso 4437  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-fv 5380  df-ov 6078  df-oprab 6079  df-mpo 6080  df-1st 6364  df-2nd 6365  df-pnf 8352  df-mnf 8353  df-xr 8354  df-ltxr 8355  df-le 8356  df-xadd 10154

This theorem is referenced by:  bldisj  15425